arXiv subjects
Publications and source records attributed to Shannon Starr.
We give a precise definition for excitations consisting of a droplet of size n in the XXZ chain with various choices of boundary conditions, including kink boundary conditions and prove that, for each n, the droplet energies converge to a boundary condition independent value in the thermodynamic limit. We rigorously compute an explicit formula for this limiting value using the Bethe ansatz.
The purpose of this article is to obtain a better understanding of the extended variational principle (EVP). The EVP is a formula for the thermodynamic pressure of a statistical mechanical system as a limit of a sequence of minimization problems. It was developed for disordered mean-field spin systems, spin systems where the underlying Hamiltonian is itself random, and whose distribution is permutation invariant. We present the EVP in the simpler setting of classical mean-field spin systems, where the Hamiltonian is non-random and symmetric. The EVP essentially solves these models. We compare the EVP with another method for mean-field spin systems: the self-consistent mean-field equations. The two approaches lead to dual convex optimization problems. This is a new connection, and it permits a generalization of the EVP.
In a recent paper we conjectured that for ferromagnetic Heisenberg models the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin and called this property ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture for the Heisenberg model with arbitrary spins and coupling constants on a chain. In this paper we give a pedagogical introduction to this result and also discuss some extensions and implications. The latter include the property that the relaxation time of symmetric simple exclusion processes on a graph for which FOEL can be proved, equals the relaxation time of a random walk on the same graph. This equality of relaxation times is known as Aldous' Conjecture.
We prove ferromagnetic ordering of energy levels for XXX Heisenberg chains of any spin and XXZ spin chains with all spins equal to 1/2. Ferromagnetic ordering means that the minimum energies in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin. This result provides a ferromagnetic analogue of the well-known theorem by Lieb and Mattis about ordering of energy levels in antiferromagnetic and ferrimagnetic systems on bipartite graphs.
We consider a notion of uniform thinning for a finite sequence of random variables $(X_1,...,X_n)$ obtained by removing one random variable, uniformly at random. If a triangular array of random variables $(X_{n,k} : n \in \mathbb{N}_+, 1 \le k \le n)$ satisfies that the law of $(X_{n,1},...,X_{n,n})$ is obtained by uniformly thinning $(X_{n+1,1},...,X_{n+1,n+1})$, then we call the array thinning-invariant. We give a representation for the Choquet simplex of all thinning-invariant triangular arrays of random variables, when all random variables take values in a compact metric space (with Borel measurable distributions). We give two applications: to long-ranged, asymmetric classical spin chains, and long-ranged, asymmetric simple exclusion processes.
We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the Lieb-Mattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest energies in each subspace of fixed total spin are strictly ordered according to the total spin, with the lowest, i.e., the ground state, belonging to the maximal total spin subspace. Our main result is a proof of this conjecture for the spin-1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof has two main ingredients. The first is an extension of a result of Koma and Nachtergaele which shows that monotonicity as a function of the total spin follows from the monotonicity of the ground state energy in each total spin subspace as a function of the length of the chain. For the second part of the proof we use the Temperley-Lieb algebra to calculate, in a suitable basis, the matrix elements of the Hamiltonian restricted to each subspace of the highest weight vectors with a given total spin. We then show that the positivity properties of these matrix elements imply the necessary monotonicity in the volume. Our method also shows that the first excited state of the XXX ferromagnet on any finite tree has one less than maximal total spin.
In \cite{Nac2,Nac1} Nachtergaele obtained explicit lower bounds for the spectral gap above many frustration free quantum spin chains by using the ``martingale method''. We present simple improvements to his main bounds which allow one to obtain a sharp lower bound for the spectral gap above the spin-1/2 ferromagnetic XXZ chain. As an illustration of the method, we also calculate a lower bound for the spectral gap of the AKLT model, which is about 1/3 the size of the expected gap.
We study the time evolution of interfaces of the ferromagnetic XXZ chain in a magnetic field. A scaling limit is introduced where the strength of the magnetic field tends to zero and the microscopic time to infinity while keeping their product constant. The leading term and its first correction are determined and further analyzed in more detail for the case of a uniform magnetic field.
We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain. Our main theorem is that there is a non-vanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about the magnitude of this gap, about its behavior for large Δ, about its overall behavior as a function of Δand its dependence on the ground state, about the scaling of the gap and the structure of the low-lying spectrum for large J, and about the existence of isolated eigenvalues in the excitation spectrum. By combining information obtained by perturbation theory, numerical, and asymptotic analysis we arrive at a number of interesting conjectures. The proof of the main theorem, as well as some of the numerical results, rely on a comparison result with a Solid-on-Solid (SOS) approximation. This SOS model itself raises interesting questions in combinatorics, and we believe it will prove useful in the study of interfaces in the XXZ model in higher dimensions.
In this thesis we present three results about the ferromagnetic quantum XXZ model: 1) Existence of a spectral gap above all infinite-volume ground states in one dimension for any choice of spin S>1/2 (for S=1/2 this was already known); 2) Characterization of one-dimensional "droplet" states as ground states of the spin-1/2 model with up spin boundary fields; 3) An upper bound for the spectral gap above the spin-1/2 interface ground states in dimension two and higher. The upper bound vanishes as 1/R^2, where R is the linear size of the system. This thesis contains research performed in collboration with other authors: O. Bolina, P. Contucci, T. Koma, B. Nachtergaele.
We consider the ground states of the ferromagnetic XXZ chain with spin up boundary conditions in sectors with a fixed number of down spins. This forces the existence of a droplet of down spins in the system. We find the exact energy and the states that describe these droplets in the limit of an infinite number of down spins. We prove that there is a gap in the spectrum above the droplet states. As the XXZ Hamiltonian has a gap above the fully magnetized ground states as well, this means that the droplet states (for sufficiently large droplets) form an isolated band. The width of this band tends to zero in the limit of infinitely large droplets. We also prove the analogous results for finite chains with periodic boundary conditions and for the infinite chain.
It is shown that, with an appropriate scaling, the energy of low-lying excitations of the (1,1,...,1) interface in the $d$-dimensional quantum Heisenberg model are given by the spectrum of the $d-1$-dimensional Laplacian on an suitable domain.
We show that the ground states of the three-dimensional XXZ Heisenberg ferromagnet with a 111 interface have excitations localized in a subvolume of linear size R with energies bounded by O(1/R^2). As part of the proof we show the equivalence of ensembles for the 111 interface states in the following sense: In the thermodynamic limit the states with fixed magnetization yield the same expectation values for gauge invariant local observables as a suitable grand canonical state with fluctuating magnetization. Here, gauge invariant means commuting with the total third component of the spin, which is a conserved quantity of the Hamiltonian. As a corollary of equivalence of ensembles we also prove the convergence of the thermodynamic limit of sequences of canonical states (i.e., with fixed magnetization).